Canad. Math. Bull. 52(2009), 39-52
Printed: Mar 2009
We present a new approach to noncommutative real algebraic geometry
based on the representation theory of $C^\ast$-algebras.
An important result in commutative real algebraic geometry is
Jacobi's representation theorem for archimedean quadratic modules
on commutative rings.
We show that this theorem is a consequence of the
Gelfand--Naimark representation theorem for commutative $C^\ast$-algebras.
A noncommutative version of Gelfand--Naimark theory was studied by
I. Fujimoto. We use his results to generalize
Jacobi's theorem to associative rings with involution.
Ordered rings with involution, $C^\ast$-algebras and their representations, noncommutative convexity theory, real algebraic geometry
16W80 - Topological and ordered rings and modules [See also 06F25, 13Jxx]
46L05 - General theory of $C^*$-algebras
46L89 - Other ``noncommutative'' mathematics based on $C^*$-algebra theory [See also 58B32, 58B34, 58J22]
14P99 - None of the above, but in this section